When digital signals are sent across a communication system, it is common to transmit the signals in the form of a modulated carrier signal. This allows the bit rate of data transfer to be increased. After transmission, the received signal must be demodulated to reconstruct the original signal (the content).
Digital modulation schemes use finite numbers of discrete signals. A predefined number of symbols (unique combinations of binary digits) may be assigned for each digital modulation scheme.
Pulse amplitude modulation (PAM) is a modulation scheme which encodes message information according to the amplitude of a series of signal pulses. Each digital symbol is encoded as a value representing a specific amplitude and polarity of the carrier signal. By assigning positive and negative polarities, positive and negative values can be assigned to specific symbols. The received signal can then be demodulated by determining which of these amplitudes the received signal is closest to, and then assigning the symbol corresponding to this amplitude. Amplitude shift keying (ASK) is a type of PAM which modulates the amplitude of a sinusoidal carrier wave.
Quadrature amplitude modulation (QAM) is a modulation scheme which encodes a digital message by modulating the amplitudes of two carrier waves using ASK. The two carrier waves are sinusoids which are out of phase with each other by 90° (a sine wave and a cosine wave). The two carrier waves are summed and can then be demodulated at the receiving end by separating out the two combined carrier waves and determining the corresponding symbol for the amplitude of each carrier wave. QAM effectively utilises two PAM signals sent in parallel.
The set of M symbols for a given modulation scheme can be represented by a constellation diagram. For a binary system, each symbol represents a unique combination of m binary digits, where m=log2 M. For QAM, each symbol can be represented by a complex number representing a position on the complex plane. The sine and cosine carrier waves are modulated by the real and imaginary parts of the complex number for a given symbol to provide the modulated signal for a given symbol. A coherent demodulator can then independently demodulate the two carrier waves to determine the symbol.
As such, demodulating the received signal may be done considering two points, each being represented in one dimension (by applying two independent PAM demodulators, one to the sine carrier wave signal and one to the cosine carrier wave signal). Embodiments described herein are therefore applicable to any separable QAM constellation arrangement.
Furthermore, where Gray codes are used to modulate the message being sent, all constellation points with the same x coordinate may have the same bit values for the first (or second) half of their Gray code identification and all constellation points with the same y coordinate may have the same bit value for the second (or first) half of their Gray code identification. As such, the consideration of the x and y axis bits can be done independently, with each considering different halves of the total set of bits for the symbol to be decoded.
From information theory it is known that, in addition to coding gain (improved bit error rate from encoding data), shaping gain can be obtained if the amplitude of the transmitter output follows a Gaussian distribution. Conventional quadrature amplitude modulation (QAM) constellations have a uniform distribution. That is, the symbols on the constellation diagram are evenly spaced in the complex plane. This means that there are even divisions between the signal amplitudes assigned to the symbols on the sine and cosine carrier waves.
Efforts have been made to design nonuniform QAM constellations which approximate the Gaussian distribution. One method samples the Gaussian Cumulative Distribution Function (CDF) in a uniform fashion to achieve this goal.
When demodulating a received (noisy) QAM signal, the log-likelihood ratio (LLR) is sought. If the received signal is y=x+v, where x is the transmitted QAM symbols and v is the noise, the LLR of the ith bit of the symbol, bi(x), is defined as
                              LLR          i                =                  log          ⁢                                    Pr              ⁡                              (                                                                            b                      i                                        ⁡                                          (                      x                      )                                                        =                                      1                    |                    y                                                  )                                                    pr              ⁡                              (                                                                            b                      i                                        ⁡                                          (                      x                      )                                                        =                                      0                    |                    y                                                  )                                                                        [        1        ]            
That is, the log likelihood ratio for a given bit in a symbol is the log of the probability of the bit being “1” divided by the probability of the bit being “0”. Assuming that the a priori probability is the same for all transmitted symbols and using the common max-log approximation, the LLR is often approximated as:
                              LLR          i                ≈                              1                          σ              2                                ⁢                      (                                                            min                                      x                    ∈                                          X                      0                                              (                        i                        )                                                                                            ⁢                                                                                                y                      -                      x                                                                            2                                            -                                                min                                      x                    ∈                                          X                      1                                              (                        i                        )                                                                                            ⁢                                                                                                y                      -                      x                                                                            2                                                      )                                              [        2        ]            where X0(i)={x|bi(x)=0}, and X1(i)={x|bi(x)=1}, and σ2 is the noise variance. That is, X0(i) and X1(i) are the set of possible QAM symbols where the ith bit equals 0 and 1 respectively. Hence the distance to all constellation points (all possible symbols) must be found and the minimum over the two subsets (X0(i) and X1(i)) must then be taken. The complexity of this is 0(M) if there are M different QAM symbols, i.e., it is proportional to the number of symbols. For uniform constellations there are efficient methods with complexity 0(log2 M) and also suboptimal methods with reduced complexity.
Known reduced complexity methods can't be used for nonuniform constellations as they assume that all symbols are equispaced. The straightforward implementation of the max-log expression requires the computation of M distances, which can be time consuming, particularly as the number of possible symbols for the modulation scheme increases.